Nonorthogonal Slater Determinants Research Articles

Automatic order detection and restoration through systematically improvable variational wave functions

Variational wave function are an invaluable tool to study the properties of strongly correlated systems. We propose such a wave function, based on the theory of auxiliary fields and combining aspects of auxiliary-field quantum Monte Carlo and modern variational optimization techniques including automatic differentiation. The resulting , consisting of several slices of optimized projectors, is highly expressive and systematically improvable. We benchmark this form on the two-dimensional Hubbard model, using both cylindrical and large, fully periodic supercells. The computed ground-state energies are competitive with the best variational results. Moreover, the optimized wave functions predict the correct ground-state order with near full symmetry restoration (i.e., translation invariance) despite initial states with incorrect orders. The can become a tool for local order prediction, leading to a new paradigm for variational studies of bulk systems. It can also be viewed as an approach to produce accurate and systematically improvable wave functions in a convenient form of nonorthogonal Slater determinants (e.g., for quantum chemistry) at polynomial computational cost. Published by the American Physical Society 2024

Generalized nonorthogonal matrix elements. II: Extension to arbitrary excitations

Electronic structure methods that exploit nonorthogonal Slater determinants face the challenge of efficiently computing nonorthogonal matrix elements. In a recent publication [H. G. A. Burton, J. Chem. Phys. 154, 144109 (2021)], I introduced a generalized extension to the nonorthogonal Wick's theorem that allows matrix elements to be derived between excited configurations from a pair of reference determinants with a singular nonorthogonal orbital overlap matrix. However, that work only provided explicit expressions for one- and two-body matrix elements between singly- or doubly-excited configurations. Here, this framework is extended to compute generalized nonorthogonal matrix elements between higher-order excitations. Pre-computing and storing intermediate values allows one- and two-body matrix elements to be evaluated with an O(1) scaling relative to the system size, and the LIBGNME computational library is introduced to achieve this in practice. These advances make the evaluation of all nonorthogonal matrix elements almost as easy as their orthogonal counterparts, facilitating a new phase of development in nonorthogonal electronic structure theory.

Multistate Density Functional Theory of Excited States.

We report a rigorous formulation of density functional theory for excited states, providing a theoretical foundation for a multistate density functional theory. We prove the existence of a Hamiltonian matrix functional of the multistate matrix density D(r) in the subspace spanned by the lowest N eigenstates. Here, D(r) is an N-dimensional matrix of state densities and transition densities. Then, a variational principle of the multistate subspace energy is established, whose minimization yields both the energies and densities of the individual N eigenstates. Furthermore, we prove that the N-dimensional matrix density D(r) can be sufficiently represented by N2 nonorthogonal Slater determinants, based on which an interacting active space is introduced for practical calculations. This work establishes that the ground and excited states can be treated on an equal footing in density functional theory.

Generalized nonorthogonal matrix elements: Unifying Wick’s theorem and the Slater–Condon rules

Matrix elements between nonorthogonal Slater determinants represent an essential component of many emerging electronic structure methods. However, evaluating nonorthogonal matrix elements is conceptually and computationally harder than their orthogonal counterparts. While several different approaches have been developed, these are predominantly derived from the first-quantized generalized Slater-Condon rules and usually require biorthogonal occupied orbitals to be computed for each matrix element. For coupling terms between nonorthogonal excited configurations, a second-quantized approach such as the nonorthogonal Wick's theorem is more desirable, but this fails when the two reference determinants have a zero many-body overlap. In this contribution, we derive an entirely generalized extension to the nonorthogonal Wick's theorem that is applicable to all pairs of determinants with nonorthogonal orbitals. Our approach creates a universal methodology for evaluating any nonorthogonal matrix element and allows Wick's theorem and the generalized Slater-Condon rules to be unified for the first time. Furthermore, we present a simple well-defined protocol for deriving arbitrary coupling terms between nonorthogonal excited configurations. In the case of overlap and one-body operators, this protocol recovers efficient formulas with reduced scaling, promising significant computational acceleration for methods that rely on such terms.

Excited-State Diffusion Monte Carlo Calculations: A Simple and Efficient Two-Determinant Ansatz.

We perform excited-state variational Monte Carlo and diffusion Monte Carlo calculations using a simple and efficient wave function ansatz. This ansatz follows the recent variation-after-response formalism, accurately approximating a configuration interaction singles wave function as a sum of only two nonorthogonal Slater determinants and further including important orbital relaxation. The ansatz is used to perform diffusion Monte Carlo calculations with large augmented basis sets, comparing to benchmarks from near-exact quantum chemical methods. The significance of orbital optimization in excited-state diffusion Monte Carlo is demonstrated, and the excited-state optimization procedure is discussed in detail. Diffuse excited states in water and formaldehyde are studied, in addition to the formaldimine and benzonitrile molecules.

Ab initio computations of molecular systems by the auxiliary‐field quantum Monte Carlo method

The auxiliary‐field quantum Monte Carlo (AFQMC) method provides a computational framework for solving the time‐independent Schrödinger equation in atoms, molecules, solids, and a variety of model systems. AFQMC has recently witnessed remarkable growth, especially as a tool for electronic structure computations in real materials. The method has demonstrated excellent accuracy across a variety of correlated electron systems. Taking the form of stochastic evolution in a manifold of nonorthogonal Slater determinants, the method resembles an ensemble of density‐functional theory (DFT) calculations in the presence of fluctuating external potentials. Its computational cost scales as a low‐power of system size, similar to the corresponding independent‐electron calculations. Highly efficient and intrinsically parallel, AFQMC is able to take full advantage of contemporary high‐performance computing platforms and numerical libraries. In this review, we provide a self‐contained introduction to the exact and constrained variants of AFQMC, with emphasis on its applications to the electronic structure of molecular systems. Representative results are presented, and theoretical foundations and implementation details of the method are discussed.This article is categorized under:Electronic Structure Theory > Ab Initio Electronic Structure MethodsStructure and Mechanism > Computational Materials ScienceComputer and Information Science > Computer Algorithms and Programming

Quantum Lattice Fluctuations in the Charge Density Wave State beyond the Adiabatic Approximation

We have developed a tractable numerical method in which large-amplitude quantum lattice fluctuations can be described beyond the adiabatic approximation using the coherent state representation of phonons. A many-body wave function is constructed by the superposition of direct products of non-orthogonal Slater determinants for electrons and coherent states of phonons. Both orbitals in all the Slater determinants and the amplitudes of all the coherent states are simultaneously optimized. We apply the method to the one-dimensional Su–Schrieffer–Heeger model with the on-site and nearest-neighbor-site Coulomb interactions. It is shown the lattice fluctuations in doped charge density wave (CDW) systems are described by the translational and vibrational motion of lattice solitons. Such lattice solitons induce bond alternation in the doped CDW system while the lattice becomes equidistant in the half-filled CDW system.

Variational description of the ground state of the repulsive two-dimensional Hubbard model in terms of nonorthogonal symmetry-projected Slater determinants

The few determinant (FED) methodology, introduced in our previous works for 1D lattices, is here adapted for the repulsive two-dimensional Hubbard model at half-filling and with finite doping fractions. Within this configuration mixing scheme, a given ground state with well defined spin and space group quantum numbers, is expanded in terms of a nonorthogonal symmetry-projected basis determined through chains of variation-after projection calculations. The results obtained for the ground state and correlation energies of half-filled and doped 4$\times$4, 6$\times$6, 8$\times$8, and 10$\times$10 lattices, as well as momentum distributions and spin-spin correlation functions in small lattices, compare well with those obtained using other state-of-the-art approximations. The structure of the intrinsic determinants resulting from the variational strategy is interpreted in terms of defects that encode information on the basic units of quantum fluctuations in the considered 2D systems. The varying nature of the underlying quantum fluctuations, reflected in a transition to a stripe regime for increasing onsite repulsions, is discussed using the intrinsic determinants belonging to a 16$\times$4 lattice with 56 electrons. Such a transition is further illustrated by computing spin-spin and charge-charge correlation functions with the corresponding multireference FED wave functions. In good agreement with previous studies, the analysis of the pairing correlation functions reveals a weak enhancement of the extended $s$-wave and d$_{x^2-y^2}$ pairing modes. Given the quality of results here reported together with those previously obtained for 1D lattices and the parallelization properties of the FED scheme, we believe that symmetry projection techniques are very well suited for building ground state wave functions of correlated electronic systems, regardless of their dimensionality.

Potential energy curves for Mo2: multi-component symmetry-projected Hartree–Fock and beyond

The molybdenum dimer is an example of a transition metal system with a formal sextuple bond that constitutes a challenging case for ab initio quantum chemistry methods. In particular, the complex binding pattern in the Mo2 molecule requires a high-quality description of non-dynamic and dynamic electron correlation in order to yield the correct shape of the potential energy curve. The present study examines the performance of a recently implemented multi-component symmetry projected Hartree–Fock (HF) approach. In this work, the spin and spatial symmetries of a trial wavefunction written in terms of non-orthogonal Slater determinants are deliberately broken and then restored in a variation-after-projection framework. The resulting symmetry-projected HF wavefunctions, which possess well-defined quantum numbers, can account for static and some dynamic correlations. A single symmetry-projected configuration in a D∞hS-UHF or a D∞hKS-UHF framework offers a reasonable description of the potential energy curve of Mo2, though the binding energy is too small for the former. Our multi-component strategy offers a way to improve on the single configuration result in a systematic way towards the exact wavefunction: in the def2-TZVP basis set considered in this study, a 7-determinant multi-component D∞hS-UHF approach yields a bond length of 2.01 Å, in good agreement with experimental results, while the predicted binding energy is 39.2 mhartree. The results of this exploratory study suggest that a multi-component symmetry-projected HF stategy is a promising alternative in a high-accuracy description of the electronic structure of challenging systems. We also present and discuss some benchmark calculations based on the CEEIS-FCI (correlation energy extrapolation by intrinsic scaling – full configuration interaction) method for selected geometries.

GPGPU Application to the Computation of Hamiltonian Matrix Elements between Non-orthogonal Slater Determinants in the Monte Carlo Shell Model

We apply the computation with a GPU accelerator to calculate Hamiltonian matrix elements between non-orthogonal Slater determinants utilized in the Monte Carlo shell model. The bottleneck of this calculation is the two-body part in the computation of Hamiltonian matrix elements. We explain an efficient computational method to overcome this bottleneck. For General-Purpose computing on the GPU (GPGPU) of this method, we propose a computational procedure to avoid the unnecessary costs of data transfer into a GPU device and aim for efficient computation with the cuBLAS interface and the OpenACC directive. As a result, we achieve about 40 times better performance in FLOPS as compared with a single-threaded process of CPU for the two-body part in the computation of Hamiltonian matrix elements.

Excited electronic states from a variational approach based on symmetry-projected Hartree–Fock configurations

Recent work from our research group has demonstrated that symmetry-projected Hartree-Fock (HF) methods provide a compact representation of molecular ground state wavefunctions based on a superposition of non-orthogonal Slater determinants. The symmetry-projected ansatz can account for static correlations in a computationally efficient way. Here we present a variational extension of this methodology applicable to excited states of the same symmetry as the ground state. Benchmark calculations on the C2 dimer with a modest basis set, which allows comparison with full configuration interaction results, indicate that this extension provides a high quality description of the low-lying spectrum for the entire dissociation profile. We apply the same methodology to obtain the full low-lying vertical excitation spectrum of formaldehyde, in good agreement with available theoretical and experimental data, as well as to a challenging model C2v insertion pathway for BeH2. The variational excited state methodology developed in this work has two remarkable traits: it is fully black-box and will be applicable to fairly large systems thanks to its mean-field computational cost.

Essentially exact ground-state calculations by superpositions of nonorthogonal Slater determinants

An essentially exact ground-state calculation algorithm for few-electron systems based on superposition of nonorthogonal Slater determinants (SDs) is described, and its convergence properties to ground states are examined. A linear combination of SDs is adopted as many-electron wave functions, and all one-electron wave functions are updated by employing linearly independent multiple correction vectors on the basis of the variational principle. The improvement of the convergence performance to the ground state given by the multi-direction search is shown through comparisons with the conventional steepest descent method. The accuracy and applicability of the proposed scheme are also demonstrated by calculations of the potential energy curves of few-electron molecular systems, compared with the conventional quantum chemistry calculation techniques.

Benchmark of the No-Core Monte Carlo Shell Model in Light Nuclei

We report the benchmark comparison of the properties in light nuclei among the Monte Carlo Shell Model, Full Configuration Interaction and No Core Full Configuration approaches. For this benchmark process, we compare results for the energies of 9 states from helium-4 to carbon-12 that includes 7 ground states and 2 excited states. The JISP16 NN interaction is adopted. The Coulomb and all other interactions are neglected. The contributions of spurious center-of-mass excitation are not discussed here. The basis space is truncated by the cutoff of the single-particle basis space, N shell. We select the optimal ħω that minimizes the energy for that state and basis space cutoff. The MCSM results are obtained by exploiting the recent development in the computation of two-body matrix elements between non-orthogonal Slater determinants and technique of the energy-variance extrapolation. All results are found to be consistent with each other to within quoted uncertainties when they could be quantified. As an exploratory attempt, we also demonstrate how to draw the intrinsic density from no-core MCSM wave functions by taking the 8Be ground state as an example.

Efficient computation of Hamiltonian matrix elements between non-orthogonal Slater determinants

We present an efficient numerical method for computing Hamiltonian matrix elements between non-orthogonal Slater determinants, focusing on the most time-consuming component of the calculation that involves a sparse array. In the usual case where many matrix elements should be calculated, this computation can be transformed into a multiplication of dense matrices. It is demonstrated that the present method based on the matrix–matrix multiplication attains ∼80% of the theoretical peak performance measured on systems equipped with modern microprocessors, a factor of 5–10 better than the normal method using indirectly indexed arrays to treat a sparse array. The reason for such different performances is discussed from the viewpoint of memory access.

Electron–electron correlation energy calculations by superposition of nonorthogonal Slater determinants

A tractable calculation algorithm to calculate essentially exact electron–electron correlation energies of few-electron systems by the superposition of nonorthogonal Slater determinants (SDs) is presented. The key to the proposed procedure for updating nonorthogonal one-electron wavefunctions to ground states is that linearly independent multiple correction functions are employed and optimized on the basis of a variational principle. The accuracy and applicability of the present scheme are demonstrated through calculations of the ground-state energies for atoms and molecules. The convergence performance to the ground state is improved by multiplying the correction functions. A drastic reduction of the number of SDs required to determine the ground-state energies and a modest increase with increasing system size compared with the full configuration interaction (CI) method are also demonstrated.

Real-space finite-difference approach for multi-body systems: path-integral renormalization group method and direct energy minimization method

The path-integral renormalization group and direct energy minimization method of practical first-principles electronic structure calculations for multi-body systems within the framework of the real-space finite-difference scheme are introduced. These two methods can handle higher dimensional systems with consideration of the correlation effect. Furthermore, they can be easily extended to the multicomponent quantum systems which contain more than two kinds of quantum particles. The key to the present methods is employing linear combinations of nonorthogonal Slater determinants (SDs) as multi-body wavefunctions. As one of the noticeable results, the same accuracy as the variational Monte Carlo method is achieved with a few SDs. This enables us to study the entire ground state consisting of electrons and nuclei without the need to use the Born–Oppenheimer approximation. Recent activities on methodological developments aiming towards practical calculations such as the implementation of auxiliary field for Coulombic interaction, the treatment of the kinetic operator in imaginary-time evolutions, the time-saving double-grid technique for bare-Coulomb atomic potentials and the optimization scheme for minimizing the total-energy functional are also introduced. As test examples, the total energy of the hydrogen molecule, the atomic configuration of the methylene and the electronic structures of two-dimensional quantum dots are calculated, and the accuracy, availability and possibility of the present methods are demonstrated.

First-principles path-integral renormalization-group method for Coulombic many-body systems

An approach for obtaining the ground state of Coulombic many-body systems is presented. This approach is based on the path-integral renormalization-group method with nonorthogonal Slater determinants, is free of the negative sign problem, and can handle higher dimensional systems with consideration of the correlation effect. Furthermore, it can be easily extended to the multicomponent quantum systems that contain more than two kinds of quantum particles. According to our results obtained with the present approach, it achieves the same accuracy as the variational Monte Carlo method with a few Slater determinants and enables us to study the entire ground state consisting of electrons and nuclei without the need to use the Born-Oppenheimer approximation.

Quantum Fluctuations due to Spinons, Polarons, and Stripes in the Two-Dimensional Hubbard Model

Quantum fluctuations (QF's) in the two-dimensional Hubbard model are visualized by superposition of optimized nonorthogonal Slater determinants. In the half-filled system, QF's consist of rotational and translational motions of spinon-antispinon pairs, while in the lightly doped systems (delta=0.96), those motions of polarons form the QF's. It is shown that an attractive interaction works between two polarons, in the framework of a projected Hartree-Fock picture. At about 10% doping, the ground state has a stripe structure and QF's due to deviations from the uniform stripe. The present method gives the ground state energies comparable or in some cases superior to the variational Monte Carlo method with a Gutzwiller projection parameter.

Quantum fluctuations in the one dimensional doped Hubbard model

Quantum fluctuations in the one dimensional (ID) doped Hubbard model are described by superposition of optimized non-orthogonal Slater determinants. It is explicitly shown that the spin and charge fluctuations are mostly separated as spinons and holons in the strong interaction regime, while the fluctuations due to the spin-charge pled states become important in the weak and intermediate interaction regimes

Visualization of quantum fluctuations by superposition of optimized nonorthogonal Slater determinants

Electronic states in the one-dimensional (1D) doped Hubbard model are described by superposition of optimized nonorthogonal Slater determinants (S-dets). Analysis on the S-dets allows us to visualize quantum fluctuations. In the weak and intermediate interaction regimes, quantum fluctuations are described by translation and breathing motions of spin-charge coupled defects called polarons as well as spin-charge decoupled defects called holons and spinons. In the strong-interaction regime, on the other hand, spin and charge fluctuations are mostly separated as spinons and holons, especially in the lightly doped systems $(\ensuremath{\delta}=0.08)$. In the highly doped systems $(\ensuremath{\delta}=0.24)$, polarons also play an important role. Finally, it is shown that in the 1/3-filled systems, the concepts of holon, spinon, and polaron do not work anymore. The electronic structure is qualitatively described by mixtures of two different density waves having triple periodicity to the lattice. The domains of these different density waves make quantum fluctuations.